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Consider the following statements: Statement 1: If lim_(x→α)(sin(f(x)))/(x-α), where f(x)=ax²+bx+c, exists as a finite nonzero number, then lim_(x→α)(1/f(x)-1)/(1/f(x)+1) does not exist. Statement 2: The limit lim_(x→α)(sin(f(x)))/(x-α) can be finite only in the indeterminate form 0/0.

1. Statement 1 is correct, Statement 2 is correct, and Statement 2 correctly explains Statement 1
2. Statement 1 is correct, Statement 2 is correct, but Statement 2 does not correctly explain Statement 1
3. Statement 1 is incorrect, Statement 2 is correct
4. Statement 1 is correct, Statement 2 is incorrect

Correct answer: Statement 1 is incorrect, Statement 2 is correct

Solution

For sin(f(x))/(x-alpha) to be finite, f(alpha)=0, so f->0 and 1/f->infinity. Then (1/f-1)/(1/f+1)->1, which exists, making Statement 1 false. Statement 2 is true since a finite limit requires the 0/0 form. Hence Statement 1 incorrect, Statement 2 correct.

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